Goppa's conjecture on the binary Gilbert–Varshamov bound

For a binary linear code $C[n,k,d]$, let $R:=\frac{k}{n}$ be its information rate and let $\delta:=\frac{d}{n}$ be its relative minimum distance. The Gilbert–Varshamov bound is the lower bound

$$\alpha_2(\delta) \geq 1-x\log_2(1)+x\log_2(x)+(1-x)\log_2(1-x),$$

where $\alpha_2(\delta)$ denotes the asymptotic achievable rate at relative minimum distance $\delta$.

Goppa's conjecture. The Gilbert–Varshamov bound is tight in the binary case; equivalently, the asymptotic achievable rate equals the Gilbert–Varshamov bound.

This conjecture asserts that the classical Gilbert–Varshamov lower bound gives the exact asymptotic rate–distance tradeoff for binary linear codes. The source provides no evidence of resolution, so its status is open.